I have taught at three colleges, and have never been at a school with a regularly-offered number theory course. As a number theorist asked to teach undergraduate abstract algebra for the first time several years ago, I was drawn to a particular textbook (Hillman & Alexanderson) by its final chapter, which was explicitly about number theory. Sivaramakrishnan’s work would be an excellent followup to that kind of algebra course.

Beginning with the premise that “it is desirable to learn algebra via number theory and to learn number theory via algebra,” this book gives a thorough treatment of both subjects and clearly shows how each illuminates the other. If a number-theoretic result has an accessible algebraic analog, it is clearly presented in its generality. At the same time, the book includes fine coverage of more elementary number theory from this advanced vantage point — the Fibonacci sequence, Fermat’s Little Theorem, and Goldbach’s conjecture are all here among the discussions of topologies, Dedekind domains, and lattices.

While not for beginners in either subject — by design, of course — this book does an excellent job of cataloging and explaining the many dimensions of the rich relationship between algebra and number theory. To read it is a great challenge, as one might expect from such a far-reaching work, but one which amply rewards careful effort.

Mark Bollman ([email protected]) is an assistant professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.